Research

Mathematics

1. [arXiv] On the equivalence of derivatives for maps between Carnot groups.

• Submitted. (2024)

2. [arXiv] [journal] Higher order Whitney extension and Lusin approximation for Horizontal curves in the Heisenberg group (with A. Pinamonti and G. Speight)

   • J. Math. Pures Appl. (2024)

3. [arXiv] [journal] Bi-Lipschitz arcs in metric spaces with controlled geometry (with J. Honeycutt and V. Vellis)

   • Rev. Mat. Iberoam. (2023)

4. [arXiv] [journal] Identifying 1-rectifiable measures in Carnot groups (with M. Badger and S. Li)

   • Anal. Geom. Metr. Spaces (2023)

5. [arXiv] [journal] A $C^{m,ω}$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group (with G. Speight)

   • J. Geo. Anal. (2023)

6. [arXiv] [journal] Singular integrals on $C^{1,α}_{w*}$ regular curves in Banach duals. 

   • Ann. Funct. Anal. (2022)

7. [arXiv] [journal] Whitney's Extension Theorem and the finiteness principle for curves in the Heisenberg group.

   • Rev. Mat. Iberoam. (2023)

8. [arXiv] [journal] Singular integrals on $C^{1,\alpha}$ regular curves in Carnot groups (with V. Chousionis and S. Li)

   • J. Anal. Math. (2021)

9. [arXiv] [journal] Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces (with V. Chousionis, S. Li, and V. Vellis)

   •  Ann. Acad. Sci. Fenn. Math. (2020)

10. [arXiv] [journal] An implicit function theorem for Lipschitz mappings into metric spaces (with P. Hajlasz) 

    • Indiana Univ. Math. J. (2020)

11. [arXiv] [journal] A $C^m$ Whitney Extension Theorem for horizontal curves in the Heisenberg group (with A. Pinamonti and G. Speight)

    • Trans Am Math Soc. (2019)

12. [arXiv] [journal] The Traveling Salesman Theorem in Carnot groups (with V. Chousionis and S. Li)

    • Calc. Var. Partial Differ. Equ. (2019)

13. [arXiv] [journal] Weak BLD mappings and Hausdorff measure (with P. Hajlasz and S. Malekzadeh)

    • Nonlinear Anal. (2018)

14. [arXiv] [journal] Sobolev extensions of Lipschitz mappings into metric spaces. 

    • Int. Math. Res. Notes IMRN. (2019)

15. [arXiv] [journal] The Whitney Extension Theorem for $C^1$, horizontal curves in the Heisenberg group. 

    • J. Geo. Anal. (2018)

16. [website] The Heisenberg groups (with P. Hajlasz)

    • Embeddings and Extrapolation (2017) 

17. [arXiv] [journal] The Dubovitskii-Sard theorem in Sobolev spaces (with P. Hajlasz)

    • Indiana Univ. Math. J. (2017) 

18. [arXiv] [journal] Geodesics in the Heisenberg group (with P. Hajlasz)

    • Anal. Geom. Metr. Spaces (2015) 

Education

1. [arXiv] [journal] Orban, Chris M.; Zimmerman, Scott; Kulp, Jessica T.; Boughton, Jennifer; Perrico, Zachary; Rapp, Brianna; Teeling-Smith, Richelle. "Methods to Simplify Object Tracking in Video Data." Phys. Teach. 1 (2023); 61 (7): 576–579.

Whitney extensions into Carnot groups

• When can we construct a smooth horizontal surface in the Heisenberg groups with prescribed derivative data on a compact subset?

• Can we make such a construction in more general Carnot groups? Which ones? What additional data is required? 

• What if we only consider curves? How regular can these curves be?

• When such a smooth horizontal surface or curve can be built, how much control do we have on the norms of the derivatives?

Quantitative rectifiability in Carnot groups

• What is the relationship between the geometry of a set in a Carnot group and the boundedness of a singular integral operator restricted to that set?

• In particular, what regularity is required for a curve in a Carnot group to guarantee the boundedness of a large class of singular integral operators on it?

• Can we classify these regular curves in terms of the bounded singular integral operators?

• To what degree can we quantify the approximation of regular sets in Carnot groups by affine sets?

BLD maps

• Can we embed any doubling metric space into a Euclidean space with a map which controls the lengths of curves?

• What kind of geometric and topological properties must such length-controlling mappings possess?

• Can we build a non-trivial winding map in the Heisenberg group? If so, how bad can it be?

• How does the branch set of such a mapping look?

Other questions

• Which results from classical analysis have a natural generalization to the setting of Lipschitz mappings from Euclidean space into metric spaces?

• Which subsets of the Heisenberg group can be embedded into a Euclidean space in a bi-Lipschiz way?

• When can a Lipschitz mapping between metric spaces be extended to a Sobolev map with a controlled norm? 

The Heisenberg Group (an intro for undergraduate math majors)

The Whitney Extension Theorem for curves in the Heisenberg group

Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean space

The Traveling Salesman Theorem in Carnot groups

Applications of a change of variables for Lipschitz mappings into metric spaces

Sobolev extensions of Lipschitz mappings into metric spaces

Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem. 

Doctoral Dissertation, University of Pittsburgh (2017). [link]